# Intro to commutative property of multiplication

CCSS Math: 3.OA.B.5
Practice changing the order of factors in a multiplication problem and see how it affects the product.

## Comparing totals

This array shows $\greenD{2}$ rows of dots with $\purpleD{4}$ dots in each row. We can use the expression $\greenD{2} \times \purpleD{4}= \goldD{8}$ to represent the array.
This array shows $\purpleD{4}$ rows of dots with $\greenD{2}$ dots in each row. We can use the expression $\purpleD{4} \times \greenD{2} = \goldD{8}$ to represent the array.
In both examples we get a total of $\goldD{8}$ dots.
$\greenD{4} \times\purpleD{2} = \goldD{8}$ and $\purpleD{2} \times \greenD{4} = \goldD{8}$
When we change the order of the numbers that we are multiplying the product stays the same.
$\greenD{5} \times \purpleD{4} = \goldD{20}$
$\purpleD{4}\times \greenD{5} = \goldD{20}$
$\greenD{5} \times \purpleD{4} = \purpleD{4}\times \greenD{5}$
$\greenD{7} \times \purpleD{10} = \goldD{70}$
$\purpleD{10}\times \greenD{7} = \goldD{70}$
$\greenD{7} \times \purpleD{10} = \purpleD{10}\times \greenD{7}$
Practice problem 1a
Match the expressions that are equal to each other.
 $3\times 9$$9 \times 4$$7 \times 4$ $4 \times 9$$9 \times 3$$4 \times 7$

We learned that $4 \times 2 = 8$.
Let's use a picture to decide if $4 + 2$ $\stackrel{?}{=}$ $8$.
We can see from the picture that $4 + 2 \neq 8$. So the answer is not the same when we change the operation.
Practice problem 1b
Which two expressions will give us the same answer?

## Commutative property

The math rule that says the order in which we multiply the factors does not change the product is the commutative property.
Let's use an array to help explain why this works. This array shows $\goldD{5}$ rows with $\blueD{2}$ dots in each row.
We can find the total number of dots by multiplying the number of rows by the number of dots in each row.
$\goldD{5} \times \blueD{2} = \greenD{ 10}$
If we turned the array on its side we have an array that shows $\blueD{2}$ rows with $\goldD{5}$ dots in each row.
All we did was tip the array over. The total number of dots did not change.
If we multiply the number of rows by the number of dots in each row we get:
$\blueD{2} \times \goldD{5} = \greenD{ 10}$
The order in which we multiply the numbers $\blueD{2}$ and $\goldD{5}$ does not matter.
$\goldD{5} \times \blueD{2} = \blueD{2} \times \goldD{5}$

### Let's try a few problems

This array shows $8$ rows with $4$ dots in each row.
Problem 2, part A
What would the array look like if we tipped it on its side?

Problem 2, part B
$8$ rows with $4$ dots $=$ $4$ rows with
dots.

Problem 2, part C
$8 \times 4 =$

## Using the commutative property

### Describing an array

The commutative property says that the order of the numbers doesn't matter in multiplication.
So the order of the numbers doesn't matter when describing an array.
We can use the expression $5 \times 3$ to show $5$ groups of $3$.
Or the expression $3 \times 5$ to show $3$ groups of $5$.
Both expressions equal $15$.

### Another problem

Practice problem 3
Which two expressions can be used to represent the array?

## Why is the commutative property helpful?

The commutative property can make multiplying more than two numbers easier.
Let's look at an example:
We can multiply $7 \times 2 \times 5$ in two steps:
$7 \times 2 = 14$
$14 \times 5 = 70$
We got the right answer, but $14 \times 5$ is a little tricky to multiply!
Remember that the commutative property lets us change the order of the numbers without changing the answer.
We can switch the $7$ and $5$ and change the problem to $5 \times 2 \times 7$. Let's see how this makes it easier to multiply:
$5 \times 2 = 10$
$10 \times 7 = 70$
Multiplying by $10$ in the second step made it easier to find the product.
Practice problem 4A
Which expressions are the same as $4 \times 3 \times 5$?
$5 \times 3 \times 6 =$
Try to rearrange the numbers so you are multiplying by a multiple of $10$.